A327595 Total number of colors in all colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors in (weakly) increasing order.

0, 1, 5, 47, 343, 2989, 33185, 360963, 4279363, 55461897, 771543693, 11345355815, 176710558327, 2913914537349, 50149603855065, 906096874764227, 17125269159665511, 336432862441344121, 6882511824853124773, 146018382159954093023, 3207861915702573763355

Offset: 0

Author

Alois P. Heinz, Sep 18 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A327244.

Programs

Maple

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(add(k*b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..21);

Mathematica

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[
    b[n-i*j, Min[n-i*j, i-1], k, p+j]/j!*c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[Sum[k*b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 11 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A327244(n,k).